In this paper, we study the p-Frechet-Urysohn property of function spaces, for p is-an-element-of beta(omega)\omega. We prove that C(pi)(X) is p-Frechet-Urysohn if and only if X has (gamma(p)), where (gamma(p) is the natural p-version of property (gamma) (this is a generalization of a result due to Gerlits and Nagy). We note the following implications: X is second countable double arrow pointing right X has (gamma(p)) for some p is-an-element-of beta(omega)\omega double line arrow pointing right X(n) is Lindelof for all 1 less-than-or-equal-to n < omega. We deal with the question when is C(pi)(R) a p-Frechet-Urysohn space. It is shown that there is p is-an-element-of beta(omega)\omega such that C(pi)(R) is p-Frechet-Urysohn

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if p is semiselective, then every subset X of R satisfying (gamma(p)) has measure zero and if p is selective, then X is a strong measure zero set

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and we can find p is-an-element-of beta(omega)\omega such that C(pi)(R) is p-Frechet-Urysohn and is not strongly p-Frechet-Urysohn. Finally, we prove that R(omega) does not have (gamma(p)) whenever p is a P-point of beta(omega)\omega.